3.67/1.71	WORST_CASE(Omega(n^1), O(n^1))
3.67/1.72	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.67/1.72	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.67/1.72	
3.67/1.72	
3.67/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.67/1.72	
3.67/1.72	(0) CpxTRS
3.67/1.72	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.67/1.72	(2) CpxTRS
3.67/1.72	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 123 ms]
3.67/1.72	    (4) BOUNDS(1, n^1)
3.67/1.72	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.67/1.72	(6) TRS for Loop Detection
3.67/1.72	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.67/1.72	    (8) BEST
3.67/1.72	        (9) proven lower bound
3.67/1.72	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.67/1.72	            (11) BOUNDS(n^1, INF)
3.67/1.72	        (12) TRS for Loop Detection
3.67/1.72	
3.67/1.72	
3.67/1.72	----------------------------------------
3.67/1.72	
3.67/1.72	(0)
3.67/1.72	Obligation:
3.67/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.67/1.72	
3.67/1.72	
3.67/1.72	The TRS R consists of the following rules:
3.67/1.72	
3.67/1.72	   not(true) -> false
3.67/1.72	   not(false) -> true
3.67/1.72	   evenodd(x, 0) -> not(evenodd(x, s(0)))
3.67/1.72	   evenodd(0, s(0)) -> false
3.67/1.72	   evenodd(s(x), s(0)) -> evenodd(x, 0)
3.67/1.72	
3.67/1.72	S is empty.
3.67/1.72	Rewrite Strategy: INNERMOST
3.67/1.72	----------------------------------------
3.67/1.72	
3.67/1.72	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.67/1.72	transformed relative TRS to TRS
3.67/1.72	----------------------------------------
3.67/1.72	
3.67/1.72	(2)
3.67/1.72	Obligation:
3.67/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.67/1.72	
3.67/1.72	
3.67/1.72	The TRS R consists of the following rules:
3.67/1.72	
3.67/1.72	   not(true) -> false
3.67/1.72	   not(false) -> true
3.67/1.72	   evenodd(x, 0) -> not(evenodd(x, s(0)))
3.67/1.72	   evenodd(0, s(0)) -> false
3.67/1.72	   evenodd(s(x), s(0)) -> evenodd(x, 0)
3.67/1.72	
3.67/1.72	S is empty.
3.67/1.72	Rewrite Strategy: INNERMOST
3.67/1.72	----------------------------------------
3.67/1.72	
3.67/1.72	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.67/1.72	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 3. 
3.67/1.72	
3.67/1.72	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.67/1.72	final states : [1, 2]
3.67/1.72	transitions: 
3.67/1.72	true0() -> 0
3.67/1.72	false0() -> 0
3.67/1.72	00() -> 0
3.67/1.72	s0(0) -> 0
3.67/1.72	not0(0) -> 1
3.67/1.72	evenodd0(0, 0) -> 2
3.67/1.72	false1() -> 1
3.67/1.72	true1() -> 1
3.67/1.72	01() -> 5
3.67/1.72	s1(5) -> 4
3.67/1.72	evenodd1(0, 4) -> 3
3.67/1.72	not1(3) -> 2
3.67/1.72	false1() -> 2
3.67/1.72	01() -> 6
3.67/1.72	evenodd1(0, 6) -> 2
3.67/1.72	02() -> 9
3.67/1.72	s2(9) -> 8
3.67/1.72	evenodd2(0, 8) -> 7
3.67/1.72	not2(7) -> 2
3.67/1.72	false1() -> 3
3.67/1.72	evenodd1(0, 6) -> 3
3.67/1.72	true2() -> 2
3.67/1.72	not2(7) -> 3
3.67/1.72	false1() -> 7
3.67/1.72	evenodd1(0, 6) -> 7
3.67/1.72	not2(7) -> 7
3.67/1.72	true2() -> 3
3.67/1.72	true2() -> 7
3.67/1.72	false2() -> 2
3.67/1.72	false3() -> 2
3.67/1.72	false3() -> 3
3.67/1.72	false3() -> 7
3.67/1.72	true3() -> 2
3.67/1.72	true3() -> 3
3.67/1.72	true3() -> 7
3.67/1.72	
3.67/1.72	----------------------------------------
3.67/1.72	
3.67/1.72	(4)
3.67/1.72	BOUNDS(1, n^1)
3.67/1.72	
3.67/1.72	----------------------------------------
3.67/1.72	
3.67/1.72	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.67/1.72	Transformed a relative TRS into a decreasing-loop problem.
3.67/1.72	----------------------------------------
3.67/1.72	
3.67/1.72	(6)
3.67/1.72	Obligation:
3.67/1.72	Analyzing the following TRS for decreasing loops:
3.67/1.72	
3.67/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.67/1.72	
3.67/1.72	
3.67/1.72	The TRS R consists of the following rules:
3.67/1.72	
3.67/1.72	   not(true) -> false
3.67/1.72	   not(false) -> true
3.67/1.72	   evenodd(x, 0) -> not(evenodd(x, s(0)))
3.67/1.72	   evenodd(0, s(0)) -> false
3.67/1.72	   evenodd(s(x), s(0)) -> evenodd(x, 0)
3.67/1.72	
3.67/1.72	S is empty.
3.67/1.72	Rewrite Strategy: INNERMOST
3.67/1.72	----------------------------------------
3.67/1.72	
3.67/1.72	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.67/1.72	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.67/1.72	
3.67/1.72	The rewrite sequence
3.67/1.72	
3.67/1.72	evenodd(s(x1_0), 0) ->^+ not(evenodd(x1_0, 0))
3.67/1.72	
3.67/1.72	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.67/1.72	
3.67/1.72	The pumping substitution is [x1_0 / s(x1_0)].
3.67/1.72	
3.67/1.72	The result substitution is [ ].
3.67/1.72	
3.67/1.72	
3.67/1.72	
3.67/1.72	
3.67/1.72	----------------------------------------
3.67/1.72	
3.67/1.72	(8)
3.67/1.72	Complex Obligation (BEST)
3.67/1.72	
3.67/1.72	----------------------------------------
3.67/1.72	
3.67/1.72	(9)
3.67/1.72	Obligation:
3.67/1.72	Proved the lower bound n^1 for the following obligation:
3.67/1.72	
3.67/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.67/1.72	
3.67/1.72	
3.67/1.72	The TRS R consists of the following rules:
3.67/1.72	
3.67/1.72	   not(true) -> false
3.67/1.72	   not(false) -> true
3.67/1.72	   evenodd(x, 0) -> not(evenodd(x, s(0)))
3.67/1.72	   evenodd(0, s(0)) -> false
3.67/1.72	   evenodd(s(x), s(0)) -> evenodd(x, 0)
3.67/1.72	
3.67/1.72	S is empty.
3.67/1.72	Rewrite Strategy: INNERMOST
3.67/1.72	----------------------------------------
3.67/1.72	
3.67/1.72	(10) LowerBoundPropagationProof (FINISHED)
3.67/1.72	Propagated lower bound.
3.67/1.72	----------------------------------------
3.67/1.72	
3.67/1.72	(11)
3.67/1.72	BOUNDS(n^1, INF)
3.67/1.72	
3.67/1.72	----------------------------------------
3.67/1.72	
3.67/1.72	(12)
3.67/1.72	Obligation:
3.67/1.72	Analyzing the following TRS for decreasing loops:
3.67/1.72	
3.67/1.72	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.67/1.72	
3.67/1.72	
3.67/1.72	The TRS R consists of the following rules:
3.67/1.72	
3.67/1.72	   not(true) -> false
3.67/1.72	   not(false) -> true
3.67/1.72	   evenodd(x, 0) -> not(evenodd(x, s(0)))
3.67/1.72	   evenodd(0, s(0)) -> false
3.67/1.72	   evenodd(s(x), s(0)) -> evenodd(x, 0)
3.67/1.72	
3.67/1.72	S is empty.
3.67/1.72	Rewrite Strategy: INNERMOST
3.67/1.74	EOF